Average Error: 30.2 → 0.5
Time: 11.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}
double f(double x) {
        double r65636 = x;
        double r65637 = 1.0;
        double r65638 = r65636 + r65637;
        double r65639 = cbrt(r65638);
        double r65640 = cbrt(r65636);
        double r65641 = r65639 - r65640;
        return r65641;
}
↓
double f(double x) {
        double r65642 = 1.0;
        double r65643 = exp(r65642);
        double r65644 = log(r65643);
        double r65645 = x;
        double r65646 = r65645 + r65642;
        double r65647 = cbrt(r65646);
        double r65648 = r65647 * r65647;
        double r65649 = cbrt(r65645);
        double r65650 = r65647 + r65649;
        double r65651 = r65650 * r65649;
        double r65652 = r65648 + r65651;
        double r65653 = r65644 / r65652;
        return r65653;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied flip3--30.1

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}}\]

  4. Simplified29.5

    \[\leadsto \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified29.5

    \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]
  6. Using strategy rm

  7. Applied add-log-exp32.1

    \[\leadsto \frac{\left(x + 1\right) - \color{blue}{\log \left(e^{x}\right)}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  8. Applied add-log-exp32.1

    \[\leadsto \frac{\left(x + \color{blue}{\log \left(e^{1}\right)}\right) - \log \left(e^{x}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  9. Applied add-log-exp32.1

    \[\leadsto \frac{\left(\color{blue}{\log \left(e^{x}\right)} + \log \left(e^{1}\right)\right) - \log \left(e^{x}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  10. Applied sum-log32.1

    \[\leadsto \frac{\color{blue}{\log \left(e^{x} \cdot e^{1}\right)} - \log \left(e^{x}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  11. Applied diff-log32.1

    \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{x} \cdot e^{1}}{e^{x}}\right)}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  12. Simplified0.5

    \[\leadsto \frac{\log \color{blue}{\left(e^{1} \cdot 1\right)}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  13. Final simplification0.5

    \[\leadsto \frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))